Scholz’s Star

19 February, 2015

100,000 years ago, some of my ancestors came out of Africa and arrived in the Middle East. 50,000 years ago, some of them reached Asia. But between those dates, about 70,000 years ago, two stars passed through the outer reaches of the Solar System, where icy comets float in dark space!

One was a tiny red dwarf called Scholz’s star. It’s only 90 times as heavy as Jupiter. Right now it’s 20 light years from us, so faint that it was discovered only in 2013, by Ralf-Dieter Scholz—an expert on nearby stars, high-velocity stars, and dwarf stars.

The other was a brown dwarf: a star so small that it doesn’t produce energy by fusion. This one is only 65 times the mass of Jupiter, and it orbits its companion at a distance of 80 AU.

(An AU, or astronomical unit, is the distance between the Earth and the Sun.)

A team of scientists has just computed that while some of my ancestors were making their way to Asia, these stars passed about 0.8 light years from our Sun. That’s not very close. But it’s close enough to penetrate the large cloud of comets surrounding the Sun: the Oort cloud.

They say this event didn’t affect the comets very much. But if it shook some comets loose from the Oort cloud, they would take about 2 million years to get here! So, they won’t arrive for a long time.

At its closest approach, Scholz’s star would have had an apparent magnitude of about 11.4. This is a bit too faint to see, even with binoculars. So, don’t look for it myths and legends!

As usual, the paper that made this discovery is expensive in journals but free on the arXiv:

• Eric E. Mamajek, Scott A. Barenfeld, Valentin D. Ivanov, Alexei Y. Kniazev, Petri Vaisanen, Yuri Beletsky, Henri M. J. Boffin, The closest known flyby of a star to the Solar System.

It must be tough being a scientist named ‘Boffin’, especially in England! Here’s a nice account of how the discovery was made:

• University of Rochester, A close call of 0.8 light years, 16 February 2015.

The brown dwarf companion to Scholz’s star is a ‘class T’ star. What does that mean? It’s pretty interesting. Let’s look at an example just 7 light years from Earth!

Brown dwarfs


Thanks to some great new telescopes, astronomers have been learning about weather on brown dwarfs! It may look like this artist’s picture. (It may not.)

Luhman 16 is a pair of brown dwarfs orbiting each other just 7 light years from us. The smaller one, Luhman 16B, is half covered by huge clouds. These clouds are hot—1200 °C—so they’re probably made of sand, iron or salts. Some of them have been seen to disappear! Why? Maybe ‘rain’ is carrying this stuff further down into the star, where it melts.

So, we’re learning more about something cool: the ‘L/T transition’.

Brown dwarfs can’t fuse ordinary hydrogen, but a lot of them fuse the isotope of hydrogen called deuterium that people use in H-bombs—at least until this runs out. The atmosphere of a hot brown dwarf is similar to that of a sunspot: it contains molecular hydrogen, carbon monoxide and water vapor. This is called a class M brown dwarf.

But as they run out of fuel, they cool down. The cooler class L brown dwarfs have clouds! But the even cooler class T brown dwarfs do not. Why not?

This is the mystery we may be starting to understand: the clouds may rain down, with material moving deeper into the star! Luhman 16B is right near the L/T transition, and we seem to be watching how the clouds can disappear as a brown dwarf cools. (Its larger companion, Luhman 16A, is firmly in class L.)

Finally, as brown dwarfs cool below 300 °C, astronomers expect that ice clouds start to form: first water ice, and eventually ammonia ice. These are the class Y brown dwarfs. Wouldn’t that be neat to see? A star with icy clouds!

Could there be life on some of these stars?

Caroline Morley regularly blogs about astronomy. If you want to know more about weather on Luhman 16B, try this:

• Caroline Morley, Swirling, patchy clouds on a teenage brown dwarf, 28 February 2012.

She doesn’t like how people call brown dwarfs “failed stars”. I agree! It’s like calling a horse a “failed giraffe”.

For more, try:

Brown dwarfs, Scholarpedia.

Earth-Like Planets Near Red Dwarf Stars

14 February, 2015

Can red dwarf stars have Earth-like planets with life?

This is an important question, at least in the long run, because 80% of the stars in the Milky Way are red dwarfs, even though none are visible to the naked eye. 20 of the 30 nearest stars are red dwarfs! It would be nice to know if they can have planets with life.

Also, red dwarf stars live a long time! They’re small—and the smaller a star is, the longer it lives. Calculations show that a red dwarf one-tenth the mass of our Sun should last for 10 trillion years!

So if life is possible on planets orbiting red dwarf stars—or if life could get there—we could someday have very, very old civilizations. That idea excites me. Imagine what a galactic civilization spanning the 80 billion red dwarfs in our galaxy could do in 10 trillion years!

(No: you can’t imagine it. You don’t have time to think of all the amazing things they could do.)

Proxima Centauri

Let’s start close to home. Proxima Centauri, the nearest star to the Sun, is a red dwarf. If we ever explore interstellar space, we may stop by this star. So, it’s worth knowing a bit about it.

We don’t know if it has planets. But it could be part of a triple star system! The closest neighboring stars, Alpha Centauri A and B, orbit each other every 80 years. One is a bit bigger than the Sun, the other a bit smaller. They orbit in a fairly eccentric ellipse. At their closest, their distance is like the distance from Saturn to the Sun. At their farthest, it’s more like the distance from Pluto to the Sun.

Proxima Centauri is fairly far from both: a quarter of a light year away. That’s about 350 times the distance from Pluto to the Sun! We’re not even sure Proxima Centauri is gravitationally bound to the other stars. If it is, its orbital period could easily exceed 500,000 years.

If Proxima Centauri had an Earth-like planet, there’s a bit of a problem: it’s a flare star.

You see, convection stirs up this star’s whole interior, unlike the Sun. Convection of charged plasma makes strong magnetic fields. Magnetic fields get tied in knots, and the energy gets released through enormous flares! They can become as large as the star itself, and get so hot that they radiate lots of X-rays.

This could be bad for life on nearby planets… especially since an Earth-like planet would have to be very close. You see, Proxima Centauri is very faint: just 0.17% the brightness of our Sun!

In fact many red dwarfs are flare stars, for the same reasons. Proxima Centauri is actually fairly tame as red dwarfs go, because it’s 4.9 billion years old. Younger ones are more lively, with bigger flares.

Proxima Centauri is just 4.24 light-years away. If explore interstellar space it may be a good place to visit. It’s actually getting closer: it’ll come within about 3 light-years of us in roughly 27,000 years, and then drift by. We should take advantage of this and go visit it soon, like in a few centuries!

Gliese 667 Cc

Gliese 667C is a red dwarf just 1.4% as bright as our Sun. Unremarkable: such stars are a dime a dozen. But it’s famous, because we know it has at least two planets, one of which is quite Earth-like!

This planet, called Gliese 667 Cc, is one of the most Earth-like ones we know today. But it’s weirdly different from our home in many ways. Its mass is 3.8 times that of Earth. It should be a bit warmer than Earth—but dimly lit as seen by our eyes, since most of the light it gets is in the infrared.

Being close to its dim red dwarf star, its year is just 28 Earth days long. But there’s something even cooler about this planet. You can see it in the NASA artist’s depiction above. The red dwarf Gliese 667C is part of a triple star system!

The largest star in this system, Gliese 667 A, is three-quarters the mass of our Sun, but only 12% as bright. It’s an orange dwarf, intermediate between a red dwarf and our Sun, which is considered a yellow dwarf.

The second largest, Gliese 667 B, is also an orange dwarf, only 5% as bright as our sun.

These two orbit each other every 42 years. The red dwarf Gliese 667 C is considerably farther away, orbiting this pair.

What could the planet Gliese 667 Cc be like?

Tidally locked planets

Since a planet needs to be close to a red dwarf to be warm enough for liquid water, such planets are likely to be be tidally locked, with one side facing their sun all the time.

For a long time, this made scientists believe the day side of such a planet would be hot and dry, with all the water locked in ice on the night side, as shown above. People call this a water-trapped world. Perhaps not so good for life!

But a new paper argues that other kinds of worlds are likely too!

In a thin ice waterworld, an ocean covers most of the planet. It’s covered with ice on the night side, maybe 10 meters thick. The day side has open ocean. Ice melts near the edge of the ice, pours into the ocean on the day side… while on the night side, water freezes onto the bottom of the ice layer.

In an ice sheet-ocean world, there’s a big ocean on the day side and a big continent on the night side. As in the water-trapped world, a lot of ice forms on the night side, up to a kilometer thick. But if there’s enough geothermal heat, and enough water, not all the water gets frozen on the night side: enough melts to form an ocean on the day side.

Needless to say, these new scenarios are exciting because they could be more conducive to life!

Read more here:

• Jun Yang, Yonggang Liu, Yongyun Hu and Dorian S. Abbot, Water trapping on tidally locked terrestrial planets requires special conditions.

Abstract: Surface liquid water is essential for standard planetary habitability. Calculations of atmospheric circulation on tidally locked planets around M stars suggest that this peculiar orbital configuration lends itself to the trapping of large amounts of water in kilometers-thick ice on the night side, potentially removing all liquid water from the day side where photosynthesis is possible. We study this problem using a global climate model including coupled atmosphere, ocean, land, and sea-ice components as well as a continental ice sheet model driven by the climate model output.

For a waterworld we find that surface winds transport sea ice toward the day side and the ocean carries heat toward the night side. As a result, night-side sea ice remains about 10 meters thick and night-side water trapping is insignificant. If a planet has large continents on its night side, they can grow ice sheets about a kilometer thick if the geothermal heat flux is similar to Earth’s or smaller. Planets with a water complement similar to Earth’s would therefore experience a large decrease in sea level when plate tectonics drives their continents onto the night side, but would not experience complete day-side dessication. Only planets with a geothermal heat flux lower than Earth’s, much of their surface covered by continents, and a surface water reservoir about 10% of Earth’s would be susceptible to complete water trapping.

From a technical viewpoint, what’s fun about this new paper is that it uses detailed climate models that have been radically hacked to deal with a red dwarf star. Paraphrasing:

We perform climate simulations with the Community Climate System Model version 3.0 (CCSM3) which was originally developed by the National Center for Atmospheric Research to study the climate of Earth. The model contains four coupled components: atmosphere, ocean, sea ice, and land. The atmosphere component calculates atmospheric circulation and parameterizes sub-grid processes such as convection, precipitation, clouds, and boundary- layer mixing. The ocean component computes ocean circulation using the hydrostatic and Boussinesq approximations. The sea-ice component predicts ice fraction, ice thickness, ice velocity, and energy exchanges between the ice and the atmosphere/ ocean. The land component calculates surface temperature, soil water content, and evaporation.

We modify CCSM3 to simulate the climate of habitable planets around M stars following Rosenbloom et al., Liu et al., and Hu & Yang. The stellar spectrum we use is a blackbody with an effective temperature of 3400 K. We employ planetary parameters typical of a super-Earth: a radius of 1.5 R, gravity of 1.38 g, and an orbital period of 37 Earth-days. The orbital period of habitable zone planets around M stars is roughly 10–100 days. We set the insolation to 866 watts per square meter and both the obliquity and eccentricity to zero. The atmospheric surface pressure is 1.0 bar, including N2, H2O, and 355 parts per million CO2.

And so on. Way cool! They consider a variety of different kinds of continents and oceans… including one where they’re just like those here on Earth—just because the data for that is easy to get!

Here’s a question I don’t know the answer to. To what extent can models like Community Climate System Model version 3.0 be tweaked to handle different planets? And what are the main things we should worry about: ways Earth-like planets can be different enough to seriously throw off the models?

We live in exciting times, where just as we’re making huge progress trying to understand the Earth’s climate in time to make wise decisions, we’re discovering hundreds of new planets with their own very different climates.

The Pentagram of Venus

4 January, 2014


This image, made by Greg Egan, shows the orbit of Venus.

Look down on the plane of the Solar System from above the Earth. Track the Earth so it always appears directly below you, but don’t turn along with it. With the passage of each year, you will see the Sun go around the Earth. As the Sun goes around the Earth 8 times, Venus goes around the Sun 13 times, and traces out the pretty curve shown here.

It’s called the pentagram of Venus, because it has 5 ‘lobes’ where Venus makes its closest approach to Earth. At each closest approach, Venus move backwards compared to its usual motion across the sky: this is called retrograde motion.

Actually, what I just said is only approximately true. The Earth orbits the Sun once every


days. Venus orbits the Sun once every


days. So, Venus orbits the Sun in

224.701 / 365.256 ≈ 0.615187

Earth years. And here’s the cool coincidence:

8/13 ≈ 0.615385

That’s pretty close! So in 8 Earth years, Venus goes around the Sun almost 13 times. Actually, it goes around 13.004 times.

During this 8-year cycle, Venus gets as close as possible to the Earth about

13 – 8 = 5

times. And each time it does, Venus moves to a new lobe of the pentagram of Venus! This new lobe is

8 – 5 = 3

steps ahead of the last one. Check to make sure:

That’s why they call it the pentagram of Venus!

When Venus gets as close as possible to us, we see it directly in front of the Sun. This is called an inferior conjunction. Astronomers have names for all of these things:

So, every 8 years there are about 5 inferior conjunctions of Venus.

Puzzle 1: Suppose the Earth orbits the Sun n times while another planet, closer to the Sun, orbits it m times. Under what conditions does the ‘generalized pentagram’ have k = mn lobes? (The pentagram of Venus has 5 = 13 – 8 lobes.)

Puzzle 2: Under what conditions does the planet move forward j = nk steps each time it reaches a new lobe? (Venus moves ahead 3 = 8 – 5 steps each time.)

Now, I’m sure you’ve noticed that these numbers:

3, 5, 8, 13

are consecutive Fibonacci numbers.

Puzzle 3: Is this just a coincidence?

As you may have heard, ratios of consecutive Fibonacci numbers give the best approximations to the golden ratio φ = (√5 – 1)/2. This number actually plays a role in celestial mechanics: the Kolmogorov–Arnol’d–Moser theorem says two systems vibrating with frequencies having a ratio equal to φ are especially stable against disruption by resonances, because this number is hard to approximate well by rationals. But the Venus/Earth period ratio 0.615187 is actually closer to the rational number 8/13 ≈ 0.615385 than φ ≈ 0.618034. So if this period ratio is trying to avoid rational numbers by being equal to φ, it’s not doing a great job!

It’s all rather tricky, because sometimes rational numbers cause destabilizing resonances, as we see in the gaps of Saturn’s rings:

whereas other times rational numbers stabilize orbits, as with the moons of Jupiter:

I’ve never understood this, and I’m afraid no amount of words will help me: I’ll need to dig into the math.

Given my fascination with rolling circles and the number 5, I can’t believe that I learned about the pentagram of Venus only recently! It’s been known at least for centuries, perhaps millennia. Here’s a figure from James Ferguson’s 1799 book Astronomy Explained Upon Sir Isaac Newton’s Principles:

Naturally, some people get too excited about all this stuff—the combination of Venus, Fibonacci numbers, the golden ratio, and a ‘pentagram’ overloads their tiny brains. Some claim the pentagram got its origin from this astronomical phenomenon. I doubt we’ll ever know. Some get excited about the fact that a Latin name for the planet Venus is Lucifer. Lucifer, pentagrams… get it?

I got the above picture from here:

Venus and the pentagram, Grand Lodge of British Columbia and Yukon.

This website is defending the Freemasons against accusations of Satanism!

On a sweeter note, the pentagram of Venus is also called the rose of Venus. You can buy a pendant in this pattern:

It’s pretty—but according to the advertisement, that’s not all! It’s also “an energetic tool that creates a harmonising field of Negative Ion around our body to support and balance our own magnetic field and aura.”

In The Da Vinci Code, someone claims that Venus traces “a perfect pentacle across the ecliptic sky every 8 years.”

But it’s not perfect! Every 8 years, Venus goes around the Sun 13.004 times. So the whole pattern keeps shifting. It makes a full turn about once every 160 years. You can see this slippage using this nice applet, especially if you crank up the speed:

• Steven Deutch, The (almost) Venus-Earth pentagram.

Also, the orbits of Earth and Venus aren’t perfect circles!

But still, it’s fun. The universe is full of mathematical beauty. It seems we need to get closer and closer to the fundamental laws of nature to make the math and the universe match more and more accurately. Maybe that’s what ‘fundamental laws’ means. But the universe is also richly packed with beautiful approximate mathematical patterns, stacked on top of each other in a dizzying way.


Talk at the SETI Institute

5 December, 2013

SETI means ‘Search for Extraterrestrial Intelligence’. I’m giving a talk at the SETI Institute on Tuesday December 17th, from noon to 1 pm. You can watch it live, watch it later on their YouTube channel, or actually go there and see it. It’s free, and you can just walk in at 189 San Bernardo Avenue in Mountain View, California, but please register if you can.

Life’s Struggle to Survive

When pondering the number of extraterrestrial civilizations, it is worth noting that even after it got started, the success of life on Earth was not a foregone conclusion. We recount some thrilling episodes from the history of our planet, some well-documented but others merely theorized: our collision with the planet Theia, the oxygen catastrophe, the snowball Earth events, the Permian-Triassic mass extinction event, the asteroid that hit Chicxulub, and more, including the global warming episode we are causing now. All of these hold lessons for what may happen on other planets.

If you know interesting things about these or other ‘close calls’, please tell me! I’m still preparing my talk, and there’s room for more fun facts. I’ll make my slides available when they’re ready.

The SETI Institute looks like an interesting place, and my host, Adrian Brown, is an expert on the poles of Mars. I’ve been fascinated about the water there, and I’ll definitely ask him about this paper:

• Adrian J. Brown, Shane Byrne, Livio L. Tornabene and Ted Roush, Louth crater: Evolution of a layered water ice mound, Icarus 196 (2008), 433–445.

Louth Crater is a fascinating place. Here’s a photo:

By the way, I’ll be in Berkeley from December 14th to 21st, except for a day trip down to Mountain View for this talk. I’ll be at the Machine Intelligence Research Institute talking to Eliezer Yudkowsky, Paul Christiano and others at a Workshop on Probability, Logic and Reflection. This invitation arose from my blog post here:

Probability theory and the undefinability of truth.

If you’re in Berkeley and you want to talk, drop me a line. I may be too busy, but I may not.

The Search For Budget-Conscious Life

18 May, 2013


Lisa and I had dinner with Gregory Benford and his wife when I visited U.C. Irvine a couple of weekends ago, and he raised an interesting point. So far, radio searches for extraterrestrial life have only seen puzzling brief signals – not long transmissions. But what if this is precisely what we should expect?

A provocative example is Sullivan, et al. (1997). This survey lasted about 2.5 hours, with 190 1.2 minute integrations. With many repeat observations, they saw nothing that did not seem manmade. However, they “recorded intriguing, non-repeatable, narrowband signals, apparently not of manmade origin and with some degree of concentration toward the galactic plane…” Similar searches also saw one-time signals, not repeated (Shostak & Tarter, 1985; Gray & Marvel, 2001 Gray, 2001). These searches had slow times to revisit or reconfirm, often days (Tarter, 2001). Overall, few searches lasted more than hour, with lagging confirmation checks (Horowitz & Sagan, 1993). Another striking example is the “WOW” signal seen at the Ohio SETI site…

That’s a quote from a paper Benford wrote with his brother and nephew:

• Gregory Benford, James Benford, and Dominic Benford, Searching for cost optimized interstellar beacons.

They claim the cheapest way a civilization could communicate to lots of planets is a pulsed, broadband, narrowly focused microwave beam that scans the sky. So, for anyone receiving this signal, there would be a lot of time between pulses. That might explain some of the above mysteries, or this one:

As an example of using cost optimized beacon analysis for SETI purposes, consider in detail the puzzling transient bursting radio source, GCRT J17445-3009, which has extremely unusual properties. It was discovered in 2002 in the direction of the Galactic Center (1.25° south of GC) at 330 MHz in a VLA observation and subsequently re-observed in 2003 and 2004 in GMRT observations (Hyman, 2005, 2006, 2007). It is a pulsed coherent source, with the ‘burst’ lasting as much as 10 minutes, with 77-minute period. Averaged over all observations, Hyman et al. give a duty cycle of 7% (1/14), although since some observations may have missed part of bursts, the duty cycle might be as high as 13%.

Even if these are red herrings, it seems very smart to figure out the cheapest ways to transmit signals and use that to guess what signals we should look for. We can easily make the mistake of assuming all extraterrestrial civilizations who bother to send signals through space will be willing to beam signals of enormous power toward us all the time. That could be true of some, but not necessarily all.

The cost analysis is here:

• James Benford, Gregory Benford, Dominic Benford, Messaging with cost optimized interstellar beacons.

and you can see a summary in this talk by Gregory’s brother James, who works on high-power microwave technologies:

The Planck Mission

22 March, 2013

Yesterday, the Planck Mission released a new map of the cosmic microwave background radiation:

380,000 years after the Big Bang, the Universe cooled down enough for protons and electrons to settle down and combine into hydrogen atoms. Protons and electrons are charged, so back when they were freely zipping around, no light could go very far without getting absorbed and then re-radiated. When they combined into neutral hydrogen atoms, the Universe soon switched to being almost transparent… as it is today. So the light emitted from that time is still visible now!

And it would look like this picture here… if you could see microwaves.

When this light was first emitted, it would have looked white to our eyes, since the temperature of the Universe was about 4000 kelvin. That’s the temperature when half the hydrogen atoms split apart into electrons and protons. 4200 kelvin looks like a fluorescent light; 2800 kelvin like an incandescent bulb, rather yellow.

But as the Universe expanded, this light got stretched out to orange, red, infrared… and finally a dim microwave glow, invisible to human eyes. The average temperature of this glow is very close to absolute zero, but it’s been measured very precisely: 2.725 kelvin.

But the temperature of the glow is not the same in every direction! There are tiny fluctuations! You can see them in this picture. The colors here span a range of ± .0002 kelvin.

These fluctuations are very important, because they were later amplified by gravity, with denser patches of gas collapsing under their own gravitational attraction (thanks in part to dark matter), and becoming even denser… eventually leading to galaxies, stars and planets, you and me.

But where did these fluctuations come from? I suspect they started life as quantum fluctuations in an originally completely homogeneous Universe. Quantum mechanics takes quite a while to explain – but in this theory a situation can be completely symmetrical, yet when you measure it, you get an asymmetrical result. The universe is then a ‘sum’ of worlds where these different results are seen. The overall universe is still symmetrical, but each observer sees just a part: an asymmetrical part.

If you take this seriously, there are other worlds where fluctuations of the cosmic microwave background radiation take all possible patterns… and form galaxies in all possible patterns. So while the universe as we see it is asymmetrical, with galaxies and stars and planets and you and me arranged in a complicated and seemingly arbitrary way, the overall universe is still symmetrical – perfectly homogeneous!

That seems very nice to me. But the great thing is, we can learn more about this, not just by chatting, but by testing theories against ever more precise measurements. The Planck Mission is a great improvement over the Wilkinson Microwave Anisotropy Probe (WMAP), which in turn was a huge improvement over the Cosmic Background Explorer (COBE):

Here is some of what they’ve learned:

• It now seems the Universe is 13.82 ± 0.05 billion years old. This is a bit higher than the previous estimate of 13.77 ± 0.06 billion years, due to the Wilkinson Microwave Anisotropy Probe.

• It now seems the rate at which the universe is expanding, known as Hubble’s constant, is 67.15 ± 1.2 kilometers per second per megaparsec. A megaparsec is roughly 3 million light-years. This is less than earlier estimates using space telescopes, such as NASA’s Spitzer and Hubble.

• It now seems the fraction of mass-energy in the Universe in the form of dark matter is 26.8%, up from 24%. Dark energy is now estimated at 68.3%, down from 71.4%. And normal matter is now estimated at 4.9%, up from 4.6%.

These cosmological parameters, and a bunch more, are estimated here:

Planck 2013 results. XVI. Cosmological parameters.

It’s amazing how we’re getting more and more accurate numbers for these basic facts about our world! But the real surprises lie elsewhere…

A lopsided universe, with a cold spot?


The Planck Mission found two big surprises in the cosmic microwave background:

• This radiation is slightly different on opposite sides of the sky! This is not due to the fact that the Earth is moving relative to the average position of galaxies. That fact does make the radiation look hotter in the direction we’re moving. But that produces a simple pattern called a ‘dipole moment’ in the temperature map. If we subtract that out, it seems there are real differences between two sides of the Universe… and they are complex, interesting, and not explained by the usual theories!

• There is a cold spot that seems too big to be caused by chance. If this is for real, it’s the largest thing in the Universe.

Could these anomalies be due to experimental errors, or errors in data analysis? I don’t know! They were already seen by the Wilkinson Microwave Anisotropy Probe; for example, here is WMAP’s picture of the cold spot:

The Planck Mission seems to be seeing them more clearly with its better measurements. Paolo Natoli, from the University of Ferrara writes:

The Planck data call our attention to these anomalies, which are now more important than ever: with data of such quality, we can no longer neglect them as mere artefacts and we must search for an explanation. The anomalies indicate that something might be missing from our current understanding of the Universe. We need to find a model where these peculiar traits are no longer anomalies but features predicted by the model itself.

For a lot more detail, see this paper:

Planck 2013 results. XXIII. Isotropy and statistics of the CMB.

(I apologize for not listing the authors on these papers, but there are hundreds!) Let me paraphrase the abstract for people who want just a little more detail:

Many of these anomalies were previously observed in the Wilkinson Microwave Anisotropy Probe data, and are now confirmed at similar levels of significance (around 3 standard deviations). However, we find little evidence for non-Gaussianity with the exception of a few statistical signatures that seem to be associated with specific anomalies. In particular, we find that the quadrupole-octopole alignment is also connected to a low observed variance of the cosmic microwave background signal. The dipolar power asymmetry is now found to persist to much smaller angular scales, and can be described in the low-frequency regime by a phenomenological dipole modulation model. Finally, it is plausible that some of these features may be reflected in the angular power spectrum of the data which shows a deficit of power on the same scales. Indeed, when the power spectra of two hemispheres defined by a preferred direction are considered separately, one shows evidence for a deficit in power, whilst its opposite contains oscillations between odd and even modes that may be related to the parity violation and phase correlations also detected in the data. Whilst these analyses represent a step forward in building an understanding of the anomalies, a satisfactory explanation based on physically motivated models is still lacking.

If you’re a scientist, your mouth should be watering now… your tongue should be hanging out! If this stuff holds up, it’s amazing, because it would call for real new physics.

I’ve heard that the difference between hemispheres might fit the simplest homogeneous but not isotropic solutions of general relativity, the Bianchi models. However, this is something one should carefully test using statistics… and I’m sure people will start doing this now.

As for the cold spot, the best explanation I can imagine is some sort of mechanism for producing fluctuations very early on… so that these fluctuations would get blown up to enormous size during the inflationary epoch, roughly between 10-36 and 10-32 seconds after the Big Bang. I don’t know what this mechanism would be!

There are also ways of trying to ‘explain away’ the cold spot, but even these seem jaw-droppingly dramatic. For example, an almost empty region 150 megaparsecs (500 million light-years) across would tend to cool down cosmic microwave background radiation coming through it. But it would still be the largest thing in the Universe! And such an unusual void would seem to beg for an explanation of its own.

Particle physics

The Planck Mission also shed a lot of light on particle physics, and especially on inflation. But, it mainly seems to have confirmed what particle physicists already suspected! This makes them rather grumpy, because these days they’re always hoping for something new, and they’re not getting it.

We can see this at Jester’s blog Résonaances, which also gives a very nice, though technical, summary of what the Planck Mission did for particle physics:

From a particle physicist’s point of view the single most interesting observable from Planck is the notorious N_{\mathrm{eff}}. This observable measures the effective number of degrees of freedom with sub-eV mass that coexisted with the photons in the plasma at the time when the CMB was formed (see e.g. my older post for more explanations). The standard model predicts N_{\mathrm{eff}} \approx 3, corresponding to the 3 active neutrinos. Some models beyond the standard model featuring sterile neutrinos, dark photons, or axions could lead to N_{\mathrm{eff}} > 3, not necessarily an integer. For a long time various experimental groups have claimed N_{\mathrm{eff}} much larger than 3, but with an error too large to blow the trumpets. Planck was supposed to sweep the floor and it did. They find

N_{\mathrm{eff}} = 3 \pm 0.5,

that is, no hint of anything interesting going on. The gurgling sound you hear behind the wall is probably your colleague working on sterile neutrinos committing a ritual suicide.

Another number of interest for particle theorists is the sum of neutrino masses. Recall that oscillation experiments tell us only about the mass differences, whereas the absolute neutrino mass scale is still unknown. Neutrino masses larger than 0.1 eV would produce an observable imprint into the CMB. [….] Planck sees no hint of neutrino masses and puts the 95% CL limit at 0.23 eV.

Literally, the most valuable Planck result is the measurement of the spectral index n_s, as it may tip the scale for the Nobel committee to finally hand out the prize for inflation. Simplest models of inflation (e.g., a scalar field φ with a φn potential slowly changing its vacuum expectation value) predicts the spectrum of primordial density fluctuations that is adiabatic (the same in all components) and Gaussian (full information is contained in the 2-point correlation function). Much as previous CMB experiments, Planck does not see any departures from that hypothesis. A more quantitative prediction of simple inflationary models is that the primordial spectrum of fluctuations is almost but not exactly scale-invariant. More precisely, the spectrum is of the form

\displaystyle{ P \sim (k/k_0)^{n_s-1} }

with n_s close to but typically slightly smaller than 1, the size of n_s being dependent on how quickly (i.e. how slowly) the inflaton field rolls down its potential. The previous result from WMAP-9,

n_s=0.972 \pm 0.013

(n_s =0.9608 \pm 0.0080 after combining with other cosmological observables) was already a strong hint of a red-tilted spectrum. The Planck result

n_s = 0.9603 \pm 0.0073

(n_s =0.9608 \pm 0.0054 after combination) pushes the departure of n_s - 1 from zero past the magic 5 sigma significance. This number can of course also be fitted in more complicated models or in alternatives to inflation, but it is nevertheless a strong support for the most trivial version of inflation.


In summary, the cosmological results from Planck are really impressive. We’re looking into a pretty wide range of complex physical phenomena occurring billions of years ago. And, at the end of the day, we’re getting a perfect description with a fairly simple model. If this is not a moment to cry out “science works bitches”, nothing is. Particle physicists, however, can find little inspiration in the Planck results. For us, what Planck has observed is by no means an almost perfect universe… it’s rather the most boring universe.

I find it hilarious to hear someone complain that the universe is “boring” on a day when astrophysicists say they’ve discovered the universe is lopsided and has a huge cold region, the largest thing ever seen by humans!

However, particle physicists seem so far rather skeptical of these exciting developments. Is this sour grapes, or are they being wisely cautious?

Time, as usual, will tell.

Black Holes and the Golden Ratio

28 February, 2013


The golden ratio shows up in the physics of black holes!

Or does it?

Most things get hotter when you put more energy into them. But systems held together by gravity often work the other way. For example, when a red giant star runs out of fuel and collapses, its energy goes down but its temperature goes up! We say these systems have a negative specific heat.

The prime example of a system held together by gravity is a black hole. Hawking showed—using calculations, not experiments—that a black hole should not be perfectly black. It should emit ‘Hawking radiation’. So it should have a very slight glow, as if it had a temperature above zero. For a black hole the mass of the Sun this temperature would be just 6 × 10-8 kelvin.

This is absurdly chilly, much colder than the microwave background radiation left over from the Big Bang. So in practice, such a black hole will absorb stuff—stars, nearby gas and dust, starlight, microwave background radiation, and so on—and grow bigger. But if we could protect it from all this stuff, and put it in a very cold box, it would slowly shrink by emitting radiation and losing energy, and thus mass. As it lost energy, its temperature would go up. The less energy it has, the hotter it gets: a negative specific heat! Eventually, as it shrinks to nothing, it should explode in a very hot blast.

But for a spinning black hole, things are more complicated. If it spins fast enough, its specific heat will be positive, like a more ordinary object.

And according to a 1989 paper by Paul Davies, the transition to positive specific heat happens at a point governed by the golden ratio! He claimed that in units where the speed of light and gravitational constant are 1, it happens when

\displaystyle{  \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2}  }

Here J is the black hole’s angular momentum, M is its mass, and

\displaystyle{ \frac{\sqrt{5} - 1}{2} = 0.6180339\dots }

is a version of the golden ratio! This is for black holes with no electric charge.

Unfortunately, this claim is false. Cesar Uliana, who just did a master’s thesis on black hole thermodynamics, pointed this out in the comments below after I posted this article.

And curiously, twelve years before writing this paper with the mistake in it, Davies wrote a paper that got the right answer to the same problem! It’s even mentioned in the abstract.

The correct constant is not the golden ratio! The correct constant is smaller:

\displaystyle{ 2 \sqrt{3} - 3 = 0.46410161513\dots }

However, Greg Egan figured out the nature of Davies’ slip, and thus discovered how the golden ratio really does show up in black hole physics… though in a more quirky and seemingly less significant way.

As usually defined, the specific heat of a rotating black hole measures the change in internal energy per change in temperature while angular momentum is held constant. But Davies looked at the change in internal energy per change in temperature while the ratio of angular momentum to mass is held constant. It’s this modified quantity that switches from positive to negative when

\displaystyle{  \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2} }

In other words:

Suppose we gradually add mass and angular momentum to a black hole while not changing the ratio of angular momentum, J, to mass, M. Then J^2/M^4 gradually drops. As this happens, the black hole’s temperature increases until

\displaystyle{ \frac{J^2}{M^4} = \frac{\sqrt{5} - 1}{2} }

in units where the speed of light and gravitational constant are 1. And then it starts dropping!

What does this mean? It’s hard to tell. It doesn’t seem very important, because it seems there’s no good physical reason for the ratio of J to M to stay constant. In particular, as a black hole shrinks by emitting Hawking radiation, this ratio goes to zero. In other words, the black hole spins down faster than it loses mass.


Discussions of black holes and the golden ratio can be found in a variety of places. Mario Livio is the author of The Golden Ratio, and also an astrophysicist, so it makes sense that he would be interested in this connection. He wrote about it here:

• Mario Livio, The golden ratio and astronomy, Huffington Post, 22 August 2012.

Marcus Chown, the main writer on cosmology for New Scientist, talked to Livio and wrote about it here:

• Marcus Chown, The golden rule, The Guardian, 15 January 2003.

Chown writes:

Perhaps the most surprising place the golden ratio crops up is in the physics of black holes, a discovery made by Paul Davies of the University of Adelaide in 1989. Black holes and other self-gravitating bodies such as the sun have a “negative specific heat”. This means they get hotter as they lose heat. Basically, loss of heat robs the gas of a body such as the sun of internal pressure, enabling gravity to squeeze it into a smaller volume. The gas then heats up, for the same reason that the air in a bicycle pump gets hot when it is squeezed.

Things are not so simple, however, for a spinning black hole, since there is an outward “centrifugal force” acting to prevent any shrinkage of the hole. The force depends on how fast the hole is spinning. It turns out that at a critical value of the spin, a black hole flips from negative to positive specific heat—that is, from growing hotter as it loses heat to growing colder. What determines the critical value? The mass of the black hole and the golden ratio!

Why is the golden ratio associated with black holes? “It’s a complete enigma,” Livio confesses.

Extremal black holes

As we’ve seen, a rotating uncharged black hole has negative specific heat whenever the angular momentum is below a certain critical value:

\displaystyle{ J < k M^2 }


\displaystyle{ k = \sqrt{2 \sqrt{3} - 3} = 0.68125003863\dots }

As J goes up to this critical value, the specific heat actually approaches -\infty! On the other hand, a rotating uncharged black hole has positive specific heat when

\displaystyle{  J > kM^2}

and as J goes down to this critical value, the specific heat approaches -\infty. So, there’s some sort of ‘phase transition’ at

\displaystyle{  J = k M^2 }

But as we make the black hole spin even faster, something very strange happens when

\displaystyle{ J > M^2 }

Then the black hole gets a naked singularity!

In other words, its singularity is no longer hidden behind an event horizon. An event horizon is an imaginary surface such that if you cross it, you’re doomed to never come back out. As far as we know, all black holes in nature have their singularities hidden behind an event horizon. But if the angular momentum were too big, this would not be true!

A black hole posed right at the brink:

\displaystyle{ J = M^2 }

is called an ‘extremal’ black hole.

Black holes in nature

Most physicists believe it’s impossible for black holes to go beyond extremality. There are lots of reasons for this. But do any black holes seen in nature get close to extremality? For example, do any spin so fast that they have positive specific heat? It seems the answer is yes!

Over on Google+, Robert Penna writes:

Nature seems to have no trouble making black holes on both sides of the phase transition. The spins of about a dozen solar mass black holes have reliable measurements. GRS1915+105 is close to J=M^2. The spin of A0620-00 is close to J=0. GRO J1655-40 has a spin sitting right at the phase transition.

The spins of astrophysical black holes are set by a competition between accretion (which tends to spin things up to J=M^2) and jet formation (which tends to drain angular momentum). I don’t know of any astrophysical process that is sensitive to the black hole phase transition.

That’s really cool, but the last part is a bit sad! The problem, I suspect, is that Hawking radiation is so pathetically weak.

But by the way, you may have heard of this recent paper—about a supermassive black hole that’s spinning super-fast:

• G. Risaliti, F. A. Harrison, K. K. Madsen, D. J. Walton, S. E. Boggs, F. E. Christensen, W. W. Craig, B. W. Grefenstette, C. J. Hailey, E. Nardini, Daniel Stern and W. W. Zhang, A rapidly spinning supermassive black hole at the centre of NGC 1365, Nature (2013), 449–451.

They estimate that this black hole has a mass about 2 million times that of our sun, and that

\displaystyle{ J \ge 0.84 \, M^2 }

with 90% confidence. If so, this is above the phase transition where it gets positive specific heat.

The nitty-gritty details

Here is where Paul Davies claimed the golden ratio shows up in black hole physics:

• Paul C. W. Davies, Thermodynamic phase transitions of Kerr-Newman black holes in de Sitter space, Classical and Quantum Gravity 6 (1989), 1909–1914.

He works out when the specific heat vanishes for rotating and/or charged black holes in a universe with a positive cosmological constant: so-called de Sitter space. The formula is pretty complicated. Then he set the cosmological constant \Lambda to zero. In this case de Sitter space flattens out to Minkowski space, and his black holes reduce to Kerr–Newman black holes: that is, rotating and/or charged black holes in an asymptotically Minkowskian spacetime. He writes:

In the limit \alpha \to 0 (that is, \Lambda \to 0), the cosmological horizon no longer exists: the solution corresponds to the case of a black hole in asymptotically flat spacetime. In this case r may be explicitly eliminated to give

(\beta + \gamma)^3 + \beta^2 -\beta - \frac{3}{4} \gamma^2  = 0.   \qquad (2.17)


\beta = a^2 / M^2

\gamma = Q^2 / M^2

and he says M is the black hole’s mass, Q is its charge and a is its angular momentum. He continues:

For \beta = 0 (i.e. a = 0) equation (2.17) has the solution \gamma = 3/4, or

\displaystyle{ Q^2 = \frac{3}{4} M^2 } \qquad  (2.18)

For \gamma = 0 (i.e. Q = 0), equation (2.17) may be solved to give \beta = (\sqrt{5} - 1)/2 or

\displaystyle{ a^2 = (\sqrt{5} - 1)M^2/2 \cong 0.62 M^2   }  \qquad  (2.19)

These were the results first reported for the black-hole case in Davies (1979).

In fact a can’t be the angular momentum, since the right condition for a phase transition should say the black hole’s angular momentum is some constant times its mass squared. I think Davies really meant to define

a = J/M

This is important beyond the level of a mere typo, because we get different concepts of specific heat depending on whether we hold J or a constant while taking certain derivatives!

In the usual definition of specific heat for rotating black holes, we hold J constant and see how the black hole’s heat energy changes with temperature. If we call this specific heat C_J, we have

\displaystyle{ C_J = T \left.\frac{\partial S}{\partial T}\right|_J }

where S is the black hole’s entropy. This specific heat C_J becomes infinite when

\displaystyle{ \frac{J^2}{M^4} = 2 \sqrt{3} - 3  }

But if instead we hold a = J/M constant, we get something else—and this what Davies did! If we call this modified concept of specific heat C_a, we have

\displaystyle{ C_a = T \left.\frac{\partial S}{\partial T}\right|_a }

This modified ‘specific heat’ C_a becomes infinite when

\displaystyle{  \frac{J^2}{M^4} = \frac{\sqrt{5}-1}{2} }

After proving these facts in the comments below, Greg Egan drew some nice graphs to explain what’s going on. Here are the curves of constant temperature as a function of the black hole’s mass M and angular momentum J:

The dashed parabola passing through the peaks of the curves of constant temperature is where C_J becomes infinite. This is where energy can be added without changing the temperature, so long as it’s added in a manner that leaves J constant.

And here are the same curves of constant temperature, along with the parabola where C_a becomes infinite:

This new dashed parabola intersects each curve of constant temperature at the point where the tangent to this curve passes through the origin: that is, where the tangent is a line of constant a=J/M. This is where energy and angular momentum can be added to the hole in a manner that leaves a constant without changing the temperature.

As mentioned, Davies correctly said when the ordinary specific heat C_J becomes infinite in another paper, eleven years earlier:

• Paul C. W. Davies, Thermodynamics of black holes, Rep. Prog. Phys. 41 (1978), 1313–1355.

You can see his answer on page 1336.

This 1978 paper, in turn, is a summary of previous work including an article from a year earlier:

• Paul C. W. Davies, The thermodynamic theory of black holes, Proc. Roy. Soc. Lond. A 353 (1977), 499–521.

And in the abstract of this earlier article, Davies wrote:

The thermodynamic theory underlying black-hole processes is developed in detail and applied to model systems. It is found that Kerr-Newman black holes undergo a phase transition at an angular-momentum mass ratio of 0.68M or an electric charge (Q) of 0.86M, where the heat capacity has an infinite discontinuity. Above the transition values the specific heat is positive, permitting isothermal equilibrium with a surrounding heat bath.

Here the number 0.68 is showing up because

\displaystyle{ \sqrt{ 2 \sqrt{3} - 3 } = 0.68125003863\dots }

The number 0.86 is showing up because

\displaystyle{ \sqrt{ \frac{3}{4} } = 0.86602540378\dots }

By the way, just in case you want to do some computations using experimental data, let me put the speed of light c and gravitational constant G back in the formulas. A rotating (uncharged) black hole is extremal when

\displaystyle{ c J = G M^2 }


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